Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fvtp2g | Structured version Visualization version GIF version |
Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Ref | Expression |
---|---|
fvtp2g | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tprot 4284 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉} | |
2 | 1 | fveq1i 6192 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) |
3 | necom 2847 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvtp1g 6463 | . . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴)) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) | |
5 | 4 | expcom 451 | . . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
6 | 5 | ancoms 469 | . . . 4 ⊢ ((𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
7 | 3, 6 | sylanb 489 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
8 | 7 | impcom 446 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) |
9 | 2, 8 | syl5eq 2668 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {ctp 4181 〈cop 4183 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: fvtp3g 6465 |
Copyright terms: Public domain | W3C validator |